Barkhausen criteria and 50% duty cycle from 555

Barkhausen criteria: An amplifier with feedback produces sustained sinusoidal oscillation at the frequency for which the loop gain Aβ satisfies both:

1. Magnitude condition: |Aβ| = 1 (loop gain is exactly unity)
2. Phase condition: the total phase shift around the loop is 0° or 360° (positive feedback), that is, ∠Aβ = 0°, 360°, ...

Here A is the amplifier gain and β is the feedback factor.

Can a 555 astable give a 50 % duty cycle?

For the 555 astable circuit, tH = 0.693 (RA + RB) C and tL = 0.693 RB C, so the duty cycle is D = (RA + RB)/(RA + 2RB). This is always more than 50 % because the capacitor charges through RA + RB but discharges only through RB. So an exact 50 % is not possible with the basic circuit. It can be obtained in these ways:

1. Make RA very small compared with RB (for example RA = 1 kΩ, RB = 100 kΩ). Then D ≈ 50 %. However, RA cannot be made too small: during the LOW time a current VCC/RA flows from VCC through RA into the discharge transistor (pin 7), which wastes power and can overload it.
2. Add a diode across RB (diode D1 from pin 7 to the capacitor node, with its cathode towards the capacitor). The capacitor then charges through RA and D1 only, and discharges through RB only:

 VCC --[RA]--+---|>|---+--- pin 6,2 --[C]-- GND
             |         |
          pin 7 ---[RB]+

tH = 0.693 RA C, tL = 0.693 RB C. Choose RA = RB to get D = 50 % (ignoring the diode drop), and f = 1/(0.693 (RA + RB) C) = 0.72/(RA C).

3. Divide the frequency by 2. Run the 555 at twice the required frequency and feed its output into a T flip-flop. The flip-flop output is an exact 50 % square wave.